The gravitational potential energy at a body of mass `m` at a distance `r` from the centre of the earth is U. What is the weight of the body at this distance ?

Options

A. `U`
B. `Ur`
C. `(U)/(r)`
D. `(U)/(2r)`

Explanation

To find the weight of a body of mass \( m \) at a distance \( r \) from the center of the Earth, given that its gravitational potential energy at that distance is \( U \), we can follow these steps:

### Step 1: Understand Gravitational Potential Energy
The gravitational potential energy \( U \) of a body at a distance \( r \) from the center of the Earth is given by the formula:
\[ U = -\frac{G M m}{r} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( r \) is the distance from the center of the Earth.

### Step 2: Relate Weight to Gravitational Force
The weight \( W \) of the body at that distance is the gravitational force acting on it, which can be expressed as:
\[ W = \frac{G M m}{r^2} \]
This formula arises from Newton's law of gravitation, where the gravitational force is inversely proportional to the square of the distance.

### Step 3: Express Weight in Terms of Potential Energy
From the expression for gravitational potential energy, we can rearrange it to find the weight:
1. We know that:
\[ U = -\frac{G M m}{r} \]
Therefore, we can express \( G M \) in terms of \( U \):
\[ G M = -\frac{U r}{m} \]
2. Substitute \( G M \) back into the weight formula:
\[ W = \frac{G M m}{r^2} = \frac{-U r}{m} \cdot \frac{m}{r^2} \]
Simplifying this gives:
\[ W = -\frac{U}{r} \]

### Step 4: Final Expression for Weight
Thus, the weight of the body at a distance \( r \) from the center of the Earth can be expressed as:
\[ W = -\frac{U}{r} \]

### Conclusion
The weight of the body at a distance \( r \) from the center of the Earth, given its gravitational potential energy \( U \), is:
\[ W = -\frac{U}{r} \]